Showing that a series diverges

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Homework Help Overview

The discussion revolves around demonstrating the divergence of the series \(\sum_{0}^{\infty}\frac{(2n)!}{(n!)^2}\left(\frac{1}{4}\right)^n\). Participants are exploring various convergence tests to apply to this series.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants express uncertainty about which convergence test to use, with some suggesting the ratio test and others questioning its applicability. There are mentions of basic tests that may not work, and a suggestion to consider Stirling's approximation.

Discussion Status

The discussion is ongoing, with participants sharing their thoughts on potential tests and clarifying the series' structure. Some guidance has been offered regarding the ratio test and the nature of the series, but no consensus has been reached on a definitive approach.

Contextual Notes

There is some confusion regarding the placement of \((1/4)^n\) within the series, which may affect the interpretation of the problem. Participants are also noting that certain tests may not be suitable for this series.

Sentral
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Homework Statement



Show that the series \sum_{0}^{\infty}(2n)!/(n!)^2*(1/4)^n diverges

Homework Equations



I don't know which convergence test to use

The Attempt at a Solution



I don't have one, because I don't know which convergence test to use. If someone can tell me what to use, I will be able to figure out this problem.
 
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Why don't you just try the most likely one?
 
I know that the basic ones such as ratio, divergence, alternating series, comparison, and integral don't work with this series. I believe it uses a test that I haven't learned about, so I was wondering what that could be.
 
I am just leaving for the evening so I don't have time to work on it myself. But I would be very surprised if the ratio test won't settle it. Did you try the root test?

I will check back later.
 
Sentral said:
I know that the basic ones such as ratio, divergence, alternating series, comparison, and integral don't work with this series. I believe it uses a test that I haven't learned about, so I was wondering what that could be.

No, one of the tests you mentioned will work. It's not an alternating series, so you can ignore that test. You wouldn't want to apply the integral test on this series, I don't think, so that eliminates that test.

Show us what you've done...
 
Clarification: Is that (1/4)^n in the numerator or the denominator of the fraction? I'm guessing the numerator, making the 4^n in the denominator, which makes the problem tougher than I thought at first glance. Is that right? I'm getting the ratio test fails too...
 
It's just the first fraction times (1/4)n. I guess there should be two parenthesis around the first fraction so it's just that quantity multiplied by the (1/4)n
 
My initial response was deleted by the moderators, but you can prove that the sum diverges by considering the sequence a_n = 1/n.
 
Last edited:
You can also learn things about series like this using Stirling's approximation.
 

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